Let $Grpd$ be the category of groupoids and $p:E\rightarrow B$ a fibration in the standard model structure on $Grpd$ (ie an isofibration). How do you prove that the pullback functor $p^{\star}:Grpd/B \rightarrow Grpd/E$ has a right adjoint ?
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Let $Grpd$ be the category of groupoids and $p:E\rightarrow B$ a fibration in the standard model structure on $Grpd$ (ie an isofibration). How do you prove that the pullback functor $p^{\star}:Grpd/B \rightarrow Grpd/E$ has a right adjoint ? Best 


Here are some references: Theorem 4.4 on p.40 of: Giraud, Jean Méthode de la descente. Bull. Soc. Math. France Mém. 2 1964 which is available from Numdam. The result was rediscovered by F. Conduché, F. Conduché, Au sujet de l'existence d'adjoints à droite aux foncteurs "image réciproque" dans la catégorie des catégories, C. R. Acad. Sci. Paris 275 (1972), A891–894. and developed for crossed complexes in Howie, James Pullback functors and crossed complexes. Cahiers Topologie Géom. Différentielle 20 (1979), no. 3, 281–296. See also Bunge, Marta; Niefield, Susan Exponentiability and single universes J. Pure Appl. Algebra 148 (2000), no. 3, 217–250. Update: I'd like to add an amusing application of the result on fibrations of groupoids and pullbacks. An epimorphism of groups, say $p: E \to B$, is a special case of a fibration of groupoids. So the pullback functor $p^*$ from groupoids over $B$ to groupoids over $E$ preserves colimits. Now the inclusion of categories $\mathbf{ Groups} \to \mathbf{ Groupoids} \;\;$ preserves colimits of connected diagrams. It follows that the pullback functor $p^*: \mathbf{ Groups}/B \to \mathbf{ Groups}/E\;\; $ preserves colimits of connected diagrams. In particular, it preserves pushouts. (This was published with P.R. Heath in "Lifting amalgamated sums and other colimits of groups and topological groups'', Math. Proc. Camb. Phil. Soc. 102 (1987) 273280.) 

